Actually, NATO took command over the last parts of Afgh july this summer, before that command was split between NATO in north and center, and US in the south. Now NATO command all of afgh - but heres the tricky part, USA is running their own show next to NATO, and does not in any way have to answer to NATO.. All in all there is alot of strange things like that going on there, such as different nations are there under different mandates an ROE's etc.. kind of an strange situation, where you end up with problems like "we have alot of high-quality troops in country, but only 3 battalions that are allowed by their governments, ROE's etc to be put in the fighting in the south etc"
Why swe and other non-NATO are there? NATO is not a closed organization. I think that in pretty much every conflict they have been involved in for the past 20 yrs or so they have welcomed any help from outside they can get. And speaking from the Swedish side, even through we are neutral, there is nothing that holds us from select certain conflicts that we feel we need to be involved whit, as long as it follows some criterias in our constitution. I think it's a similar situation for other non-NATO nations in Afgh and other conflicts as well.
Re the transport issue. Im not fully sure exactly how it works, but iirc NATO has a collective transport fleet that is distributed between the different members through a way complicated system, that in the end means that who ever needs transport capabilities wont get it, cuz the other nations that dont need it have it for the moment . As a response to that the EU is planning to build up their own transport abilities by buying a few of this C5 whatever huge planes from the US.. and come up with a new system to distribute transport measures to whoever doesent need it for the moment heh..